This study is concerned with the topography of nanostructures consisting of arrays of poly-crystalline nanodots. Guided by transmission electron microscopy (TEM) measurements of c-Si nanodots that evolved from a "dewetting" process of an a-Si layer from a SiO$_2$ coated substrate, we investigate appropriate surface energy density formulations to model these equilibrium geometries. We explore the influence of smooth transitions between the energy density states at grain boundaries on the associated surface morphology of the equilibrium poly-crystals. Furthermore, we introduce a new numerical minimization formulation that allows to account for adhesive energy from an underlying substrate. We demonstrate our approach first for the unbounded case, where the solutions can be compared to well-known Wulff constructions, before we treat the general case for interfacial energy settings that support partial 'wetting'. Eventually, we use the method to study two-dimensional shapes of poly-crystalline silicon nanodots.